Sky Sports said United will pay 51 million euros ($56.70 million) for the 27-year-old plus four million in add-ons.
United had been looking for a new first-choice keeper after Spaniard David De Gea's contract expired in June, bringing an end to his 12-season tenure with the Old Trafford side.
Onana joined Inter in July 2022 from Ajax Amsterdam and quickly became first-choice between the sticks and a central figure in a campaign that ended with Inter winning the Italian Cup and Super Cup as well as reaching the Champions League final.